Covariance Estimation for Matrix-variate Data via Fixed-rank Core Covariance Geometry

We study the geometry of the fixed-rank core covariance manifold arising from the Kronecker-core decomposition of covariance matrices. As shown in Hoff, McCormack, and Zhang (2023), every covariance matrix $\Sigma$ of $p_1\times p_2$ matrix-variate data uniquely decomposes into a separable component $K$ and a core component $C$. Such a decomposition also exists for rank-$r$ $\Sigma$ if $p_1/p_2+p_2/p_1<r$, with $C$ sharing the same rank. If this core $C$ exhibits a partial-isotropy structure, then a partial-isotropy rank-$r$ core is a non-trivial convex combination of a rank-$r$ core and $I_p$ for $p:=p_1p_2$, where the weight on $I_p$ measures the deviation of $\Sigma$ from separability. This motivates studying the geometry of the space of rank-$r$ cores, $\mathcal{C}_{p_1,p_2,r}^+$. We show that $\mathcal{C}_{p_1,p_2,r}^+$ is a smooth manifold, except for a measure-zero subset associated with canonical decomposability. When $r=p$, $\mathcal{C}_{p_1,p_2}^{++}:=\mathcal{C}_{p_1,p_2,p}^+$ is itself a smooth manifold. The geometric properties, including smoothness of the positive definite cone via separability and the Riemannian gradient and Hessian operator relevant to $\mathcal{C}_{p_1,p_2,r}^+$, are also derived. As an application, we propose a partial-isotropy core shrinkage estimator for matrix-variate data.